Space-efficient classical and quantum algorithms for the shortest vector problem

TL;DR

This paper introduces space-efficient classical and quantum algorithms for the Shortest Vector Problem in lattices, significantly reducing space requirements while maintaining competitive time complexity, advancing cryptographic applications.

Contribution

The paper presents the first space-efficient classical and quantum algorithms for SVP, reducing space complexity without sacrificing much on time complexity.

Findings

Classical algorithm solves SVP in 2^{2.05n+o(n)} time with 2^{0.5n+o(n)} space.

Quantum algorithm solves SVP in 2^{1.2553n+o(n)} time with 2^{0.5n+o(n)} classical space.

Quantum algorithm uses only poly(n) qubits.

Abstract

A lattice is the integer span of some linearly independent vectors. Lattice problems have many significant applications in coding theory and cryptographic systems for their conjectured hardness. The Shortest Vector Problem (SVP), which is to find the shortest non-zero vector in a lattice, is one of the well-known problems that are believed to be hard to solve, even with a quantum computer. In this paper we propose space-efficient classical and quantum algorithms for solving SVP. Currently the best time-efficient algorithm for solving SVP takes $2^{n+o(n)}$ time and $2^{n+o(n)}$ space. Our classical algorithm takes $2^{2.05n+o(n)}$ time to solve SVP with only $2^{0.5n+o(n)}$ space. We then modify our classical algorithm to a quantum version, which can solve SVP in time $2^{1.2553n+o(n)}$ with $2^{0.5n+o(n)}$ classical space and only poly(n) qubits.